🌸 MATH 285 - Intro Differential Equations
Math
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This deck contains everything taught in UIUC's MATH 285 - Intro Differential Equations course.
In the future, I will update this deck such that it includes material from MIT's 18.03SC course as well, so be sure to check back for that!
The MATH 285 course is based on the textbook Differential Equations by Bronski & Manfroi (UIUC) and
supplementarily based on Elementary Differential Equations and Boundary Value Problems, 10th Ed by Boyce & DiPrima
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Sample Data
| Text | The solution to the {{c1::one-dimension wave equation::name of equation}} {{c1::\(\begin{align}\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u^2}{\partial x^2 }\end{align}\)::equation}} with {{c2::homogeneous Dirichlet boundary conditions \(u(0,t)=0,\quad u(L,t)=0\)::what boundary conditions}} is\(u(x,t)\) \(=\) \(\begin{align}\sum_{n=1}^\infty \,c_n \sin\left(\frac{n \pi x}L\right)\end{align}\) \(\begin{align}\cos \left(\frac{n\pi \textcolor{MediumTurquoise}ct}{L}\right)\end{align}\) \(+\:k_n\:\begin{align}\sin\left(\frac{n \pi x}L\right)\end{align}\) \(\begin{align}\sin \left(\frac{n\pi \textcolor{MediumTurquoise}c t}{L}\right)\end{align}\) where {{c3::the quantities \(\Large \frac{2L}n\) for \(n\) \(=\) \(1,2,\ldots\)::what}} are the {{c4::wavelengths of the fundamental modes::name}} of the elastic string. |
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| Summary 1 | Derivation |
| Back Extra | The wavelength is the "spatial period," the period of oscillations in the \(x\)-direction.The spatial angular frequency \(\omega = \Large \frac{n\pi}L\) is found as the coefficient of \(x\) in \(\cos(\omega x)\) and \(\sin(\omega x)\).The spatial period \(T={\Large \frac{2\pi}{\omega} } ={\Large \frac{2\pi}{n\pi/L} } ={\Large \frac{2L}{n} } \) |
| Summary 2 | How to remember the wavelength, easily |
| Back Extra 2 | The wavelength for the fundamental mode \(n=1\) is \(2L\), then every increase in \(n\) divides up that wavelength into \(n\) smaller pieces,so you divide \(2L\) by \(n\)
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| Summary 3 | Related values |
| Back Extra 3 | The eigenfunction solutions \(u_n(x,t)\) for \(n\) \(=\) \(1,2,\ldots\) are the natural modes of vibration of the elastic string.The quantities \(\Large \frac{n\pi c}L\) for \(n\) \(=\) \(1,2,\ldots\) are the natural frequencies of the fundamental modes of the elastic string. |
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| Text | {{c1::\(y''''(t)+p_3(t)y'''(t)+p_2(t)y''(t)+p_1(t)y'(t) +p_0(t)y(t)=g(t)\)::linear differential equation}}The above linear differential equation can be written as the system of first order linear equations: {{c2::\[\begin{cases}y'(t)=z_1(t)\\ z_1'(t)=z_2(t)\quad(=y''(t))\\ z_2'(t)=z_3(t)\quad(=y'''(t))\\ z_3'(t)=-p_0(t)y(t)-p_1(t)z_1(t)-p_2(t)z_2(t)-p_3(t)z_3(t)+g(t)\end{cases}\]::system of equations}} which can also be written as {{c3::\(\vec{\mathbf v}' = \mathbf P(t) \vec{\mathbf v}+\mathbf{\vec g}(t)\)\(\begin{pmatrix}\textcolor{white}{y'(t)} \\ \textcolor{white}{z_1'(t)}\\\textcolor{white}{z_2'(t) }\\\textcolor{white}{z_3'(t) }\end{pmatrix}=\begin{bmatrix}0&\textcolor{white}1&0&0\\ 0 & 0 & \textcolor{white}1 & 0\\ 0 & 0 & 0 & \textcolor{white}1\\ \textcolor{white}{-p_0(t)}&\textcolor{white}{-p_1(t)}&\textcolor{white}{-p_2(t)}&\textcolor{white}{-p_3(t)} \end{bmatrix}\begin{pmatrix}\textcolor{white}{y(t)}\\ \textcolor{white}{z_1(t)}\\ \textcolor{white}{z_2(t)}\\ \textcolor{white}{z_3(t)}\end{pmatrix}+\begin{pmatrix}0\\0\\0\\ \textcolor{white}{g(t)}\end{pmatrix}\)::matrix form}}{{c4::}} |
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| Summary 1 | Rewritten with only \(y\) |
| Back Extra | Notice the pattern.This can be rewritten as\(\begin{pmatrix}\textcolor{white}{y'} \\ \textcolor{white}{y''}\\\textcolor{white}{y''' }\\\textcolor{white}{y'''' }\end{pmatrix}=\begin{bmatrix}0&\textcolor{white}1&0&0\\ 0 & 0 & \textcolor{white}1 & 0\\ 0 & 0 & 0 & \textcolor{white}1\\ \textcolor{white}{-p_0(t)}&\textcolor{white}{-p_1(t)}&\textcolor{white}{-p_2(t)}&\textcolor{white}{-p_3(t)} \end{bmatrix}\begin{pmatrix}\textcolor{white}y\\ \textcolor{white}{y'}\\ \textcolor{white}{y''}\\ \textcolor{white}{y'''}\end{pmatrix}+\begin{pmatrix}0\\0\\0\\ \textcolor{white}{g(t)}\end{pmatrix}\) |
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| Back Extra 2 | Given the complementary solution to \(\vec{\mathbf v}' = \mathbf P(t) \vec{\mathbf v}\), we can use variation of parameters to solve for the particular solution,then the general solution is the sum of the complementary solution and the particular solution. |
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| Back Extra 3 | If \(\mathbf P(t)\) is only made up of constants (which aren't dependent on \(t\)), then the fundamental matrix is \(\mathbf M(t)=\begin{pmatrix}|& |& &|\\\vec{\mathbf v}^{(1)}&\vec{\mathbf v}^{(2)}&\cdots&\vec{\mathbf v}^{(n)}\\|& |& &|\\\end{pmatrix}=e^{\mathbf P t}\)where the set of vector-valued functions \(\vec{\mathbf v}^{(1)}(t)\textcolor{white}, \vec{\mathbf v}^{(2)}(t)\textcolor{white}{,\ldots,}\vec{\mathbf v}^{(n)}(t)\) to \(\vec{\mathbf v }'(t) =\mathbf P(t)\vec{\mathbf v}(t)\) are the fundamental solutions. |
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| Text | A/an {{c1::nonlinear differential equation::type of differential equation}} of order \(n\) can not be written of the form \[a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+\cdots +a_n(t)y=g(t)\] |
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| Back Extra | For example, for a function \(y(t)\):\(\cos(t)t^2\,y''-e^t\,y'=t^3+4\ln t\quad\) is linear.\(ty'-t^3+2^t=y'''+\sin(t)\,y''\quad\) is linear.\((y'')^2+y'=ty-1\quad\) is nonlinear.\(y''\cdot y'=y+1\quad\) is nonlinear. |
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